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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 425031, 10 pages
Pullback Attractors for Nonautonomous 2D-Navier-Stokes Models with Variable Delays
School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China
Received 5 January 2013; Accepted 30 March 2013
Academic Editor: de Dai
Copyright © 2013 Xiaoli Liu and Yejuan Wang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Using a method based on the concept of the Kuratowski measure of the noncompactness of a bounded set as well as some new estimates of the equicontinuity of the solutions, we prove the existence of a unique pullback attractor in higher regularity space for the multivalued process associated with the nonautonomous 2D-Navier-Stokes model with delays and without the uniqueness of solutions.
It is well known that the Navier-Stokes equations are very important in the understanding of fluids motion and turbulence. These equations have been studied extensively over the last decades (see [1–3], and the references cited therein). Recently, Caraballo and Real  considered global attractors for functional Navier-Stokes models with the uniqueness of solutions and for the delay, so that a wide range of hereditary characteristics (constant or variable delay, distributed delay, etc.) can be treated in a unified way. Very recently, Marín-Rubio and Real  used the theory of multivalued dynamical system to establish the existence of attractors for the 2D-Navier-Stokes model with delays, when the forcing term containing the delay is sublinear and only continuous.
For the study of asymptotic behavior for functional partial differential equations without the uniqueness of solutions, as far as we know, not many papers have been published. However, some results in the finite dimensional context can be found in [6, 7] (see also [8–10] for some preliminary and interesting results on the structure of the attractors for ordinary differential delay systems).
The pullback attractor is a possible approach to define an “attractor” for the nonautonomous dynamical systems, the long time behavior of nonautonomous dynamical systems is an interesting and challenging problem; see, for example, [11–19], and so forth. The purpose of our current paper is to study existence of pullback attractors for the following functional Navier-Stokes problem: where is an open bounded set with regular boundary , is the kinematic viscosity, is the velocity field of the fluid, is the pressure, is the initial time, is a nondelayed external force field, is another external force term and contains some memory effects during a fixed interval of time of length , is an adequate given delay function, and the initial datum on the interval .
Using the technique of measure of noncompactness, noting that all norms on finite dimensional spaces are equivalent, we apply the new method to check the pullback -limit compactness given in  and then get the existence of the pullback attractors in .
We consider the following usual abstract spaces: where the closure of in with norm and inner product , where for , where the closure of in with norm and associated scalar product , where for , Note that , where the injections are dense and compact. We will use for the norm in and for the duality pairing between and .
Define the trilinear form on by
Now, let us establish some assumptions for (1).
We assume that the given delay function satisfies , and there exists a constant satisfying Furthermore, we suppose that and satisfy the following assumptions:(H1) is measurable for all ,(H2) is continuous for all ,(H3) there exist positive constants , such that for any , (H4) there exists a fixed such that for any , the external force satisfies
Set as , by , for all . Denote by the corresponding orthogonal projection . We further set . The Stokes operator is self-adjoint and positive from to . The inverse operator is compact. Excluding the pressure, the system (1) can be written in the form
Let be a complete metric space with metric , and denote by the class of nonempty subsets of . As usual, let us denote by the Hausdorff semidistance between and , which are defined by where . Finally, denote by the open neighborhood of radius of a subset of a Banach space .
Definition 1. A family of mappings is called to be a multivalued process (MVP in short) if it satisfies(1)for all ;(2)for all .
Let be a nonempty class of parameterized sets .
Definition 2. Let be a multivalued process on . One says that is(1) pullback -dissipative, if there exists a family , so that for any and each , there exists a such that (2) pullback -limit-set compact with respect to each , if for any and , there exists a such that where is the Kuratowski measure of noncompactness.
Definition 3. A family of nonempty compact subsets is called to be a pullback -attractor for the multivalued process , if it satisfies(1) is invariant; that is, (2) is pullback -attracting; that is, for every and any fixed ,
Let , be two Banach spaces, and let , be their dual spaces, respectively. We also assume that is a dense subspace of , the injection is continuous, and its adjoint is densely injective.
Theorem 4 (see [21, 22]). Let be two Banach spaces satisfy the previous assumptions, and let be a multivalued process on and , respectively. Assume that is upper semicontinuous or weak upper semicontinuous on . If for fixed , maps compact subsets of into bounded subsets of , then is norm-to-weak upper semicontinuous on .
By slightly modifying the arguments of Theorem 3.4 and Remark 3.9 in , we have the following.
Theorem 5. Let be a Banach space, and let be a multivalued process on . Also let be norm-to-weak upper semicontinuous in for fixed , ; that is, if , then for any , there exist a subsequence and a such that (weak convergence). Then the multivalued process possesses a pullback -attractor in given by if and only if is pullback -dissipative and pullback -limit-set compact with respect to each , where is pullback -absorbing for the multivalued process .
A multivalued process is said to be pullback -asymptotically upper-semicompact in if for each fixed , any , any sequence with , with , and any with ; this last sequence is relatively compact in .
Remark 6. Let be a multivalued process on . Then is pullback -asymptotically upper-semicompact if and only if is pullback -limit-set compact; see .
Let be a Banach space, and let be a given positive number (the delay time). Denote by the Banach space endowed with the norm Let us consider a class of sets parameterized in time, . To study the pullback -limit-set compactness of the multivalued process on , we need the following result from .
Theorem 7. Let be a multivalued process on . Suppose that for each , any and , there exist , a finite dimensional subspace of , and a such that(1) for each fixed , (2) for all , , with , (3) for all , , where is the canonical projector. Then is pullback -limit-set compact in with respect to each .
3. Existence of an Absorbing Family of Sets in
By the classical Faedo-Galerkin scheme and compactness method, analogous to the arguments in , we have the following.
Theorem 8. Let one consider , , and assume that satisfies the hypotheses (H1)–(H3). Then, for each ,(a) there exists a weak solution to problem (9) satisfying (b) if , then there exists a strong solution to problem (9); that is,
Given and , for each , we denote by the function defined on by the relation , . We also denote and . Let be the arbitrary positive constants, which may be different from line to line and even in the same line.
Thanks to Theorem 8, we can define a multivalued process as
We first need a priori estimates for the solution of (9) in the space and a necessary bound on the term , which will be very useful in our analysis; it relates the absorption property for the multivalued process on .
Lemma 9. In addition to the assumptions (H1)–(H4), assume that holds true. Then provided that is small enough.
Proof. By the energy inequality and the Poincaré inequality, we have
We fixed two positive parameters and to be chosen later on. Then by and Young’s inequality, we can deduce that
Let to be determined later on. Then it follows that
Integrating between and , we have
Let ; note that and for all . Hence, Combining (30) and (31) together, we get Let and using (23), so we can choose positive constants and small enough such that and (where is given in the assumption ). Then, it follows that Setting now instead of (where ), multiplying by , it holds Note that , thus the conclusion (24) follows immediately from (34).
Finally, we will obtain the bound on the term . It follows from (28) that Integrating from to , we have Similar to the arguments of (31), we can deduce that Recall that and . By (24) and (36)-(37), we have (25) as desired, and thus the proof of this lemma is completed.
By slightly modifying the proof of Lemma 1.1 in , we have the following result.
Lemma 10. Let be given arbitrarily. Let , , and be three positive locally integrable functions on such that is locally integrable on , which satisfy that where , , and are positive constants. Then
Now we state and prove the main result in this section.
Theorem 11. Suppose in addition to the hypotheses in Lemma 9, assume that holds true. Then the multivalued process on is pullback -dissipative.
Proof. We take the inner product of (9) with , we obtain
Now we evaluate the terms, using and Young’s inequality, and we arrive to
Thanks to (41)–(43) and the fact that for , we can deduce that
Since and , it is easy to see that . Then
Let be given arbitrarily and taking such that . In order to apply Lemma 10, in view of (24), now we firstly obtain Then, it follows from (24) and (25) that Combining (25) and (47)-(48) together, by Lemma 10, we can conclude that where Therefore, if we take such that , then similar to the above mentioned, we get
We denote by the set of all functions such that and denote by the class of all families such that , for some , where denotes the family of all nonempty subsets of and denotes the closed ball in centered at zero with radius .
Denote by the nonnegative number given for each by and consider the family of closed balls in defined by It is straightforward to check that , and moreover, by (51) and (52), the family of is pullback -absorbing for the multivalued process on .
The proof of Theorem 11 is completed.
4. Existence of the Pullback Attractors in
Theorem 12. Suppose in addition to the hypotheses in Theorem 11 that . Then there exists a unique pullback -attractor for the multivalued process in .
Proof. Since is a continuous compact operator in , by the classical spectral theory, there exist a sequence ,
and a family of elements of which are orthonormal in such that
Let in and be an orthogonal projector.
Let , where and . We decompose (9) as follows: We divide the proof into three steps.
(1) For every fixed , any and , we observe that for any with , Taking the inner product in of (57) with , we get By and Young’s inequality, we have To estimate , we recall some inequalities : and thus Note that , and set . Then by Young’s inequality, we can deduce that By (60)–(64) and Poincaré inequality, we obtain Applying the Gronwall’s lemma in the interval , it yieldsLet be given arbitrarily. Note that , then we can take large enough such that for any fixed , Combining (67) and (68) together, we can get for large enough, On the other hand, thanks to Lemma 9 and Theorem 11, we can deduce that when and are large enough, Thanks to (69) and (70), it follows from (66) that when and are large enough,
(2) Now we consider the ordinary functional differential system (58) and check the condition (2) in Theorem 7. Note that . Without generality, we assume that with . Hence Notice that Then, it follows from , , and (24) that Since and is fixed, Equations (74)–(75) imply that the condition (2) in Theorem 7 is proved.
(3) Invoking Theorem 7, in view of the previous arguments and Theorem 11, we can see that the multivalued process is pullback -limit-set compact and pullback -dissipative in .
In order to get the existence of pullback -attractors, by the proof of Theorem 3.2 in , now we only need to show the negative invariance of , where and is a pullback -absorbing set of in .
Let . Then there exist sequences ,, , and such that On the other hand, for sufficiently large, Then by the pullback -limit-set compactness of the multivalued process , there is a subsequence of , which we still relabel as such that and Clearly, .
We observe that is bounded in for sufficiently large. Then by slightly modifying the proof of the existence of solutions (see  for details), in view of Theorem 2.11 in , we can see that This together with (77)–(79), we can deduce that , and thus the proof of Theorem 12 is finished.
This research was supported by the National Natural Science Foundation of China under Grant no. 10801066 and the Fundamental Research Funds for the Central Universities under Grant no. lzujbky-2011-47 and no. lzujbky-2012-k26. The Project was sponsored by the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry.
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